A Multivariable Adaptive Control Design with Applications to Air-heat Tunnel Using Delta Models

Veiw figure View Figure

The transfer matrix of the system is

(7)

The matrix is a rational matrix (its elements are rational functions). Every rational matrix can be expressed using a polynomial matrix in the form of the left or right matrix fraction. It is possible to assume that the system is described by the matrix fraction

(8)

Polynomial matrices A, B are left indivisible decomposition of matrix , polynomial matrices A₁, B₁ are right indivisible decomposition of matrix .

The matrices of discrete model take following form

(9)

(10)

and the differential equations of the model are

(11)

3.1. Recursive Identification

In the case of the above described system with two inputs and two outputs it is necessary to identify a total of sixteen unknown parameters of ARX model described by equation (11). The unknown parameters of the - model are estimated using recursive least squares method (RLSM) with directional forgetting.

The parameter vector takes the form:

(12)

The data vector is

(13)

where

(14)

The detailed description of recursive identification algorithm for TITO system is designed in ^[8].

4. Designing of Feedback MIMO System

Transfer function of the controller takes the form matrix fraction.

(15)

Polynomial matrices P, Q are left indivisible decomposition of matrix , polynomial matrices P₁, Q₁, are right indivisible decomposition of matrix .

Block diagram of closed loop can be seen in Figure 2.

Figure 2. Block diagram of the closed loop system

Download as

Veiw figure View Figure

To ensure permanent zero control error is necessary to include the matrix of integrator. The matrix of integrator takes the form

(16)

The control law can be derived from block diagram

(17)

It is possible to derive following equation for the system output

(18)

Equation (19) can be modified to give

(19)

The stability control system is ensured if the transfer controller matrix is given by the solution matrix diophantine equation

(20)

where M is a stable diagonal polynomial matrix. The behavior of closed loop system is given by the roots of this polynomial matrix.

(21)

If the system is to be stable the roots of this polynomial matrix must be inside the circle with center at point -1/T₀ with radius 1/T₀.

The degree of the controller matrix polynomials depends on the internal properness of the closed loop. The structure of matrices P₁ and Q₁ was chosen so that the number of unknown controller parameters equals the number of algebraic equations resulting from the solution of the diophantine equations using the uncertain coefficients method.

(22)

(23)

The solution to the diophantine equation results in a two sets of eight algebraic equations with unknown controller parameters. The controller parameters are given by solving these equations. These sets can be rewrite in a matrix form.

5. Autonomous Control Using Compensators

Multiple-input multiple-output systems describe processes with more than one input and more than one output which require multiple control loops. These systems can be complicated through loop interactions that result in variables with unexpected effects. Loop interactions need to be avoided because changes in one loop might cause destabilizing changes in another loop. Decoupling the variables of that system will improve the control of that process.

There are several ways to control multivariable systems with internal interactions. Some make use of decentralized PID controllers, whilst others are composed of a string of single input – single output methods ^{[2, 4]}.

One possibility is the serial insertion of a compensator ahead of the system ^{[8, 11, 12]}. The aim here is to suppress of undesirable interactions between the input and output variables so that each input affects only one controlled variable.

Figure 3. Closed loop system with compensator

Download as

Veiw figure View Figure

The resulting transfer function is then given by

(24)

Compensator inserted in series before the system is chosen so that the product of the matrices was diagonal matrix. When matrix H is diagonal the decoupling conditions are fulfilled.

Several well – known compensators are given in ^{[8, 11, 12]}. Control algorithms were derived for the model above with two compensators. These will be referred to as C₁and C₂. Compensator C₁is based on finding the inversion of the controlled system. Matrix H is, therefore, a unit matrix.

Figure 4. Closed loop systems with compensator C₁

Download as

Veiw figure View Figure

System output takes the form

(25)

The stability of the closed loop is given by solution of following diophantine equation

(26)

The structure of polynomial matrices of controller

(27)

(28)

and matrix was chosen to be

(29)

The controller parameters are given by solution of diophantine equation (29) using the uncertain coefficients method. The control law is described by matrix equation

(30)

Compensator C₂is adjugated to matrix B. When C₂ was included in the design of the closed loop the model was simplified by considering matrix A as diagonal. The multiplication of matrix B and adjugated matrix B results in diagonal matrix H. The determinants of matrix B represent the diagonal elements. When matrix is non-diagonal, its inverted form must be placed ahead of the system in order to obtain diagonal matrix H, otherwise it may increase the order of the controller and sophistication of the closed loop system. Although designed for a diagonal matrix, compensator C₂also improves the control process for non – diagonal matrix A in the controlled system. This is demonstrated in the simulation results.

Figure 5. Closed loop systems with compensator C₂

Download as

Veiw figure View Figure

Equation for system output takes the form

(31)

The matrix is

(32)

The stability of the closed loop is given by solution of following diophantine equation

(33)

The structure of the matrix P1 and Q1 is chosen so that the number of algebraic equations after multiplication of matrix diophantine equation corresponds to the number of unknown parameters. The structure of controller polynomial matrices takes the form

(34)

(35)

Polynomial matrix M takes the form selected with regard to the structure of other matrices in diophantine equation

(36)

Solving the diophantine equation defines a set of algebraic equations. These equations are subsequently used to obtain the unknown controller parameters.

The control law is given by the block diagram

(37)

6. Simulation Examples

A program and diagrams to simulate and verify the algorithms was created in the program system MATLAB - SIMULINK. Verification by simulation was carried out on a range of systems with varying dynamics. The control of the model below is given here as our example.

(38)

(39)

The right side control matrices are denoted as follows: without compensator – M₁, with compensator C₁-M₂, and with compensator C₂-M₃.

(40)

(41)

(42)

The same initial conditions for system identification were used for all the types of adaptive control we tested. The initial parameter estimates were chosen to be

(43)

The simulation results are shown in Figure 6-8.

It is possible to draw several conclusions from the simulation results of the experiments on linear static systems. The basic requirement to ensure permanent zero control error was satisfied in all cases. The criteria on which we judge the quality of the control process are the overshoot on the controlled values and the speed with which zero control error is achieved. According to these criteria the controller incorporating compensator C₁ performed the best. However, this controller appears to be unsuited to adaptive control due to the size of the overshoot and the large numbers of process and controller outputs. The controller which uses compensator C2 seems to work best in adaptive control. The addition of compensators in series ahead of the system caused that change in one of control variables change only the corresponding process variable in all cases. Compensators actually provide autonomous control loop. With regards to decoupling, it is clear that controllers with compensators greatly reduce interaction.

7. Laboratory Experiment

The verification of the proposed algorithms for autonomous and non-autonomous adaptive multivariable control on the real object under laboratory conditions has been realized using experimental laboratory model – air-heat tunnel. It is a suitable tool for the laboratory experimental verification of control algorithms and controller parameter settings.

Figure 6. Simulation results: adaptive control without compensator

Download as

Veiw figure View Figure

Figure 7. Simulation results: adaptive control without compensator C₁

Download as

Veiw figure View Figure

Figure 8. Simulation results: adaptive control without compensator C₂

Download as

Veiw figure View Figure

The model is composed of the heating coils, primary and secondary ventilator and a thermal resistor covered by tunnel. The heating coils are powered by controllable source of voltage and serves as the source of heat energy while the purpose of ventilators is to ensure and measure the flow of air inside the tunnel. Connecting the real model - hot-air tunnel is made using a technology card Advantech PCL 812, which is connected to the motherboard.

Figure 9. Scheme of hot-air tunnel

Download as

Veiw figure View Figure

Figure 10. Adaptive control of a real model without compensator

Download as

Veiw figure View Figure

Figure 11. Adaptive control of a real model using compensator C₂

Download as

Veiw figure View Figure